Rotary screw machine and method of transforming a motion in such a machine

ABSTRACT

The invention relates to a rotary screw machine of volume type comprising a body ( 30 ) having a main axis X, two members ( 10, 20 ), wherein a first one ( 20 ) surrounds a second one ( 10 ). Said first member ( 20 ) is hinged in said body ( 30 ) and is able to swivel on itself about its axis (Xf), aligned with said main axis X, according to a swiveling motion, whereas the axis (Xm) of said second member ( 10 ), revolves about the axis of said first member (Xf) according to an revolution motion having said length E as a radius. The machine further comprises a synchronizer ( 34, 36, 38, 40 ) synchronizing said swiveling motion and said revolution motion, such that a working medium performs a volumetric displacement in at least one working chamber ( 11 ) delimited by an outer surface ( 22 ) of said first member ( 20 ) and a inner surface ( 12 ) of said second member ( 10 ).

One aspect of the invention relates to a rotary screw machine of volumetype comprising a body, two members consisting of a male member and afemale member surrounding said male member, wherein an outer surface ofthe male member defines a male surface and a inner surface of the femalemember defines a female surface, said male and female surfaces beinghelical surfaces having respective axes Xm and Xf that are parallel andspaced apart by a length E, said male and female surfaces defining atleast one working chamber by formation of linear contacts of said maleand female surfaces and relative displacement of said male and femalemembers, said male and female surfaces being further defined about saidaxes Xm and Xf by a nominal profile in a cross section of the mechanism,said profile of the male surface defining a male profile having an orderof symmetry Nm with respect to a center Om located on said male axis Xm,said profile of the female surface defining a female profile having anorder of symmetry Nf with respect to a center Of located on said femaleaxis Xf, said rotary screw machine further comprising a crank likemechanism generating an eccentricity E between said main axis X and oneof the axis Xm or Xf.

Such a rotary screw machine of volume type is known for transformingenergy of a working substance (medium), gas or liquid, by expanding,displacing and compressing said working medium, into mechanical energyfor engines or vice versa for compressors, pumps, etc.

Such a rotary screw machine of three-dimensional type is known from U.S.Pat. No. 5,439,359, wherein a male member surrounded by a fixed femaleorgan is in planetary motion relative to the female member and whereinthe outer surface of the male member defines a male surface and an innersurface of the female member defines a female surface, said male andfemale having parallel axis spaced apart by a length E (eccentricity).

A first component of this planetary motion drives the axis of the malesurface to make this axis describe a cylinder of revolution having aradius E about the axis of the female surface, which corresponds to anorbital revolution motion.

A second component of this planetary motion drives the male member tomake it rotate about the axis of its male surface. This second component(peripheral rotation), will in all the following text be calledswiveling motion.

This known rotary screw machine has only two degrees of freedom and onlyone of them is independent, e.g. if an independent degree of freedom isthe first component, orbital revolution of the male member, then thedependent degree of freedom is the swiveling motion of the male member,since the latter is guided in its swiveling motion by the contactsbetween the male and female surfaces, and vice versa.

Consequently, this rotary screw machine has limited technical potentialand has significant heat losses.

It is an object of the present invention to provide a rotary screwmachine in which technical and functional potential are broader, inreducing the angular extent of thermodynamic cycles, improvingefficiency, and in which the overall heat losses are decreased.

The invention provides a rotary screw machine in which a first one ofthe male and female members is hinged in the body and is able to rotateon itself about its fixed axis according to a rotation motion, in whichthe crank organ is connected (hinged) to a second one of the male andfemale members to allow the axis of the second member to revolve aboutfixed the axis of the first member according to an orbital revolutionmotion having the length E as a radius, and which comprises asynchronizer for synchronizing the swiveling motion and the orbitalrevolution motion, one with respect to the other, so that the male andfemale surfaces mesh together.

In all the text, when the axis of a member moves in a circular orbitaround a fixed axis of another member, it will be specified as torevolve an axis, and the process of the orbital rotation of a memberaxis in a circle around a fixed axis of another member, it will bespecified as revolution.

In the process of revolution, when a movable member rotates about itsown axis moving in orbit, it will be specified as to swivel a member,and the process itself of a peripheral rotation of a member about itsown axis moving in orbit, it will be specified as swiveling.

Thus, the planetary motion represents the sum of revolution andswiveling. When swiveling is equal to zero and revolution is not equalto zero, then the planetary motion becomes a circular progressivemotion.

The crank organ and the first one of the male and female members can beindependently controlled leading to the independence of the rotationmotion and the orbital revolution motion.

Thus, the rotary screw machine has two independent degrees of freedom.According to a preferred embodiment, the rotary screw machine furthercomprises a one-channel rotational transmission means connected to saidcrank organ or to said first member or a two-channel rotationaltransmission means connected to the crank organ and to the first member.

In this case, the crank organ and the first member are driven togetherwith the rotational transmission means and with independent choice ofmotion speeds.

In a preferred embodiment, the male and female surfaces are brought inmechanical contact forming a kinematic pair allowing the transmission ofmotion between the first and second members.

Such a rotary screw machine has three degrees of freedom two of thembeing independent, which introduces an additional rotation motion of thefirst member. The axis of the second member is able to revolve about theaxis of the first member and the second member itself is able to swivelabout its movable axis due to the self-meshing of the male and femalesurfaces, which leads to a planetary motion of the second memberrelative to the first member axis, the first member itself being able torotate about its fixed axis.

In particular, when the number of forming arcs of the female is higherthan the forming arcs of the male profile, then synchronization isprovided by self-meshing of the elements, i.e. without specialsynchronizing mechanisms.

According to a preferred embodiment, when mechanical contacts areundesirable or not easy to obtain or just to improve the drive of thesecond member, the rotary screw machine further comprises an additionalsynchronizer, linked to the body and allowing the second member toswivel about its axis.

According to the type of additional synchronizer, for example aplanetary gear, the swiveling motion speed of the second member isproportional (preferably increased, that is with a coefficient ofproportionality greater than one) to the swiveling motion speed of thefirst member.

According to a preferred embodiment, the rotary screw machine furthercomprises rotational transmission means connected to the crank organ andto one of the male or female members.

The first and second members being both in rotation and swivelingmotion, the rotation transmission means can be connected either with thefirst and/or the second member and/or crank according to the specificarrangement of the elements composing the rotary screw machine. Thus,the first member can be driven by the second member, which is then thedriving member and which is itself connected to the rotationaltransmission means and vice versa.

In a preferred embodiment, the synchronizer further comprises akinematical coupling mechanism of both members together, the kinematicalcoupling mechanism comprising at least one coupling organ, which ishinged in the body.

Thus, the both crank organ and the driving member, else the crank organor the driving member can be driven by the rotational transmissionmeans, so that their motions can be equal or different relative to eachother. The relation between their motions is given by the type ofcoupling organs chosen.

In a preferred embodiment, the kinematical coupling mechanism comprisesa planetary gear whose disposition between the crank organ and thedriving member can lead to a multiplication or a reduction of theelement being driven by the planetary gear relative to the elementconnected to the rotational transmission means.

In a preferred embodiment, the synchronizer comprises a planetary geartransmission, or an inverter or a coulisse mechanism.

The inverter is used to inverse the way of the rotation motion of thesecond member axis relative to the rotation motion of the first member.According to the disposition of the planetary gear relation with thesecond member, both preceding motions can occur in the same direction orin an opposite direction. Thus, the inverter can be used either inaddition or substitution of the planetary gear transmission.

The efficiency of the rotary screw machine being proportional to thespeed of the cycles consisting in opening and closing the chambersdefined between the first and second surfaces, it is all the highersince both first and second members are in motion. However, the bestresult is obtained when the rotation motion speed of the first member isequal to the revolution motion speed of the second member axis, butoccurs in the opposite direction of rotation. In this case, themechanical strengths applied by the first and second members against thebody are equal and opposite, such that the resultant momentum ispractically nil. These kinds of machines are used in cases where thevibrations are to be avoided or greatly limited. Generally, two or morerotating elements of rotary screw machines (including contra-rotatingelements) can be coupled through transfer mechanisms to rotatingelements of outer units or mechanisms. The coupling of this type can becarried out, e.g. in combined operation of contra-rotating volumemachine in the mode of engine with outer contra-rotor devices such ascontra-rotor turbine, contra-rotor compressor or contra-rotor electricalmachine, contra-rotor wings of air or sea vehicles, contra-rotor cuttingtools etc.

The efficiency of the rotary screw machine can also be improved inincreasing the number of first and second members.

Thus, according to a preferred embodiment, the rotary screw machinefurther comprises either at least one additional male and female membersdisposed in line with the said male and female members, or at least athird member disposed inside or surrounding the male and female members,in such a way that their surfaces are in mechanical contact so as toform additional chambers.

In a preferred embodiment, the female order of symmetry Nf is equal toNm−1, or Nm+1.

To make the realization easier of both male and female members, they canbe done as an assembly of a plurality of identical members having ad hocnominal profile and being oriented relative to each other so as todefine at least one working chamber that extends axially. The angulardistance between two consecutive elements is directly linked to thenumber of elements chosen.

When the number of elements is finite, the working medium with which themachine exchanges energy can be admitted via a cross section at one endof the mechanism and can escape via its other end.

In a preferred embodiment, the male and female surfaces can degenerateinto cylindrical surfaces.

Another aspect of the invention relates to a method of transforming amotion in a volume screw machine.

The invention relates to a method of transforming a motion in a volumescrew machine with inner conjugation of screw members with a positivedisplacement of volumes of working chambers of three-dimensional (3-D)type, which are formed by a conjugated enclosing (female) and enclosed(male) screw members.

Methods of transforming a motion are used for conversing a mechanicalenergy of a motion and working substance energy in working chambers of ascrew machine, and for transmitting a positive energy flow ofconversion. It is significant that conversion and transmission of apositive energy flow of conversion is a reversible process. The methodsare based on the creation of interconnected relative motions ofsynchronizing coupling links and the screw conjugated male and femalemembers, which form with their inner and outer helicoidal surfaces theworking chambers moving axially in the process of transforming a motion.

The known methods of transforming a motion in volume screw machinesunder conversion of a positive energy comprise: transmission of positiveenergy flow of conversion through a kinematics channel of a mechanicalrotation formed by the independent degree of freedom of the membersexecuting a planetary motion, driving one of male or female members intoplanetary motion with two degrees of freedom of mechanical rotation, ofwhich one being an independent degree of freedom relative to the fixedcentral axis of the other member.

On one hand, an outer envelope of the male profile can be an initialtrochoid of symmetry order Nm, then the internally conjugated femaleprofile presents an outer envelope of a family of trochoids of symmetryorder Nf=Nm+1 and both profiles have constantly Nm+1 points of contact.

On the other hand, an outer envelope of the male profile can be made asan inner envelope of a trochoid family mentioned above of symmetry orderNm, and the female profile is, in this case, a trochoid of symmetryorder Nf=Nm−1 and both profiles have constantly Nm points of contact.

In both cases, the contact points are kinks of one of the envelopes andmake possible to insulate constantly the working chambers via thecontacts between female and male surfaces. The inner female surface andouter male surface are screw surfaces with parallel axes, some of themcan be movable and spaced at a distance, which we denote as theeccentricity E.

In the known methods of transforming a motion in volume screw machinesthe coordinated motion of the members with the pitches (periods) Pm andPf of twist of the rated profiles of the end sections of the members isexecuted. The initial twist is performed in a pair of conjugated membersin the planes, which are normal to the longitudinal principal axis ofthe screw members, and is a birotative process of a turn of the endsections about their central axis. Relationship of the pitches of thefemale and male surfaces is determined by relation of the symmetryorders of mentioned profiles according to$\frac{Pf}{Pm} = {\frac{{Nm} + 1}{Nm}.}$In the known machines with an inner envelope, the quantity of theworking chambers are equal Nm, and an axial pitch of each workingchamber is equal Pm, whereas in the known machines with an outerenvelope, the quantity of the working chambers are equal Nm+1, and anaxial pitch of each working chamber is equal Pf.

At the finite values of Pm and Pf, in the process of transforming amotion of the members with the help of synchronizing coupling links (orby self-synchronization in the machines with an outer envelope), it ispossible to set in a planetary motion of any one of the members (male orfemale) with respect to the other (fixed) member with two degrees offreedom, one of which being an independent degree of freedom of amechanical rotation.

All known methods of transforming a motion in volume screw machines ofinner conjugation amount to the next two methods: rotary (more oftencalled as birotative) and planetary methods.

According to the first method a rotation (rotation of a member about itsown fixed axis) in one direction about a fixed parallel axis, isimparted simultaneously to the interconnected rotation of the twolinks—female and male members with the initial and conjugated screwprofiles.

According to the second one, the planetary motion is imparted to onemember (it is technically preferable to impart the planetary motion tomale member), so that its center is moved in a circle around the centerof the second member, in this case, the fixed member (female member).

Generally, with the help of synchronizing coupling links (or byself-synchronization in the machines with an outer envelope), it ispossible to set in a planetary motion of any one of the members (male orfemale) with respect to the other fixed member, with the two degrees offreedom one of which being independent.

In the known methods, a fixed female member generally sets the malemember in a planetary motion relative to the fixed central axis of thefemale member and surround it.

As it was shown above, a planetary motion can be represented as a sum oftwo components of the rotations—revolution and swiveling. The firstcomponent of rotation of this planetary motion makes the axis of themale surface describe a cylinder with a radius E about the central axisof the female fixed surface, herewith an axis of the planetary memberrevolves in orbit of radius E at an arbitrary speed ω. The secondcomponent of this planetary motion is swiveling, i.e. a peripheralrotation of the male member about its movable axis at the${speed} \pm \frac{\omega}{Nm}$(minus—when the male member is trochoidal, plus—when the male member isan inner envelope).

Effectiveness of the method of transforming a motion in the particularscrew machine is determined by intensity of the thermodynamic processestaking place in the machine, and is characterized by the generalizedparameter “angular cycle”. The cycle is equal to a turn angle of anyrotating member (male, female or synchronizing link) chosen as a memberwith an independent degree of freedom.

In the known methods, performing a function of the kinematics channel ofadmission and escape of positive energy of conversion can be an outputshaft of synchronizing link, e.g. a crank shaft of the male member andso on.

The angular cycle is equal to a turn angle of a member with independentdegree of freedom at which an overall period of variation of the crosssection area (or overall opening and closing) of the working chamber,formed by the male and female members takes place, as well as axialmovement of the working chamber by one period Pm in the machines with aninner envelope or by one period Pf in the machines with an outerenvelope.

On transforming a planetary motion of a female member, made as an outerenvelope, revolution of male member axis can be chosen as an independentrotation and swiveling of the male member is a dependent rotation. Thenthe angular cycle is defined by the angle of revolution of the malemember's axis, which is equal to$\gamma = {\frac{\pi\quad{Nm}}{{Nm} - 1}.}$This angle is equal the turn angle of a crankshaft of the synchronizinglink (with which the male member, hinged on the crank, executing theswiveling motion in the process of a planetary motion) and when apositive mechanical energy is admitted through the kinematicscrank-channel with an independent degree of freedom.

On admitting a positive energy of mechanical rotation directly to a malemember, the swiveling motion of the male member is chosen as theindependent rotation, and the revolution of the male member axis as adependent one. Swiveling of the male member with independent degree offreedom about its own movable axis through self-synchronizingconjugation of male and female members causes an axis revolution(dependent degree of freedom) in an orbit with E radius about a fixedaxis of the female member. The angular cycle in this case is equal to$\gamma = {\frac{\pi}{{Nm} - 1}.}$

The known methods of transforming a motion are used in particular indownhole motors in petroleum, gas or geothermal drilling (such asdescribed in French Patent FR-A-99 7957 and U.S. Pat. No. 3,975,120).

The transformation of a motion used in motors is described by V.Tiraspolskyi (“Hydraulical Downhole Motors in Drilling”, the course ofdrilling, pp.258-259, Published in Edition, Technip, Paris 15e). Similartransformation of a motion in those motors is carried out usually atfixed female member, which is a female member, while the planetarymotion of the male member relative to this female member is accordinglyidentified by its absolute motion.

The known methods of transforming a motion in volume screw machines withconjugated elements of a curvilinear shape realized in the similarvolume machines have the following drawbacks:

limited technical potential, because of imperfect process of organizinga motion, which fails to increase a quantity of angular cycles per oneturn of the drive member with the independent degree of freedom;

limited specific power of similar screw machines;

limited efficiency;

existence of reactive forces on the fixed body of the machine.

The invention is intended to solve a problem of widening technical andfunctional potential capabilities of the method of transforming a motionin screw machines by creating an additional kinematics channel forpositive energy of conversion with the independent degree of freedom ofa motion, i.e. by increasing the total quantity of degrees of freedom ofrotary motion up to the three, of which two of them are independent. Itprovides an increase in the efficiency of the method, an increase inquantity of angular cycles of volume change of the displacing chambersper one turn of a drive shaft and, as a result of which, tointensification of conversion processes of positive energy and decrease(up to zero) in the mechanical reactive forces on the supports of thevolume screw machine.

According to the second aspect of the invention, the second independentdegree of freedom of rotary motion is introduced in transforming amotion of male and female members and links of synchronizing coupling.On transforming a planetary motion the member, an axis of which is incoincidence with a central fixed axis, is actuated into a rotary motionabout the fixed axis with independent degree of freedom of a rotarymotion. For this purpose a portion of the positive energy of conversionis transmitted through the second independent degree of freedom ofmechanical rotation of the member executing a rotary motion aboutcentral fixed axis. -In the method according to the invention, thedifferential interconnected rotary motions of a link of synchronizingcoupling and male and female members are executed. Any two rotations ofsaid three ones (rotation, revolution and swiveling) are chosen asindependent degrees of freedom of rotary motion and the third rotationis a dependent differential function of the two independent rotations,herewith the revolution of the axis of a planetary element about centralfixed axis at radius E is created simultaneously with swiveling of thiselement and with a rotation of another conjugated element about itscentral fixed axis.

A method of transforming a motion in a volume screw machine according tothe invention, comprises the creation of interconnected motions of thescrew conjugated elements in the form of male and female members andlinks of synchronizing coupling with the help of converted positiveflows of mechanical energy and working substance energy in workingchambers of said volume screw machine, driving one of male or femalemember into a planetary motion with two degrees of freedom of mechanicalrotation one of which being an independent degree of freedom, thetransmission of said positive energy flow of conversion through anindependent degree of freedom of a mechanical rotation of said machine.

In a preferred embodiment, the method provides the creation of adifferentially connected motion of male and female members and links ofsynchronizing coupling with the second independent degree of freedom ofa rotary motion and the transmission of positive energy flow ofconversion in the form of the two flows through the two independentdegrees of freedom of a mechanical rotation of said machine.

Furthermore, according to another embodiment, at least, one dependentdegree of freedom of rotary motion can be created in the process oftransforming a motion of male and female members and links ofsynchronizing coupling, and a part of positive energy flow of conversioninside said machine can be used in transforming a motion through anadditional dependent degree of freedom of mechanical rotation of saidmachine with decreasing the number of independent degrees of freedom perunity.

According to another embodiment, the angular velocities of said memberscan be determined as differentially connected to one another accordingto the relation:k ₁ω₁ +k ₂ω₂+ω₃=0,where: ω₁,ω₂ represent the angular speed of the said conjugated elementsabout their axis;

ω₃ represents the angular speed of the link of synchronizing coupling;

k₁,k₂ represent the constant coupling coefficients; herewith, values ofangular velocities of rotation of conjugated elements are defined fromrelation:(z−1)ω₁ −zω ₂+ω₀=0,where: ω₁ represents is the angular speed of the member around its axis,enveloping surface of which has the form of curvilinear surface;

ω₂ represents the angular speed of rotation of the member around itsaxis, enveloping surface of which has a shape of inner or outer envelopeof a family of surfaces, formed with the said curvilinear surface;

ω₀ represents the angular speed of the orbital revolution of the axis ofthe member executing planetary motion;

z represents an integer, z>1.

Furthermore, according to another embodiment of the method, any two ofthe three rotations can be synchronized between one another, namely, therotation of one of the conjugated elements about their fixed axis, therevolution of an axis of the element performing a planetary motion withthe link of synchronizing coupling and the swiveling of the element witha movable axis.

The rotary screw machine of the present invention will be more fullyunderstood with reference to the accompanying figures that shownon-limiting examples.

FIG. 1 shows a longitudinal section of a rotary screw volume machineembodied with rotational motion of female member and circularprogressive motion of the male member with an inner envelope, in whichNf=Nm−1,

FIG. 2 is a cross section on the line II-II of FIG. 1,

FIG. 3 shows a longitudinal section of the rotary screw volume machineembodied with rotational motion of female member and circularprogressive motion of the male member with an outer envelope, in whichNf=Nm+1,

FIG. 4 is a cross section on the line IV-IV of FIG. 3,

FIG. 5 shows a longitudinal section of the screw volume machine embodiedwith rotation of female member with an outer envelope, in which Nf=Nm+1and circular progressive motion of the male member,

FIG. 6 is a cross section on the line VI-VI of FIG. 5,

FIG. 7 shows a longitudinal section of another embodiment of a rotaryscrew volume machine with rotational motion of male member and circularprogressive motion of the female member, in which Nf=Nm−1,

FIG. 8 is a cross section on the line VIII-VIII of FIG. 7,

FIG. 9 shows a longitudinal section of a contra-rotating screw volumemachine with two-channel rotational transmission means and withplanetary motion of male member and rotational motion of the femalemember, in which Nf=Nm−1,

FIG. 10 is a cross section on the line X-X of FIG. 9,

FIG. 11 shows a longitudinal section of a contra-rotating rotary screwvolume machine with one-channel rotational transmission means and withplanetary motion of male member and rotational motion of the femalemember, in which Nf=Nm−1,

FIG. 12 is a cross section on the line XII-XII of FIG. 11,

FIG. 13 shows a longitudinal section of a contra-rotating screw volumemachine with one independent degree of rotation of the female member, inwhich Nf=Nm−1,

FIG. 14 is a cross section on the line XIV-XIV of FIG. 13,

FIG. 15 shows a longitudinal section of a contra-rotating screw volumemachine with two independent degrees of revolution of crank passingthrough male axis and rotation of the female member in which Nf=Nm+1,

FIG. 16 is a cross section on the line XVI-XVI of FIG. 15,

FIG. 17 shows a longitudinal section of a contra-rotating screw volumemachine with planetary motion of male member and rotational motion ofthe female member, in which Nf=Nm+1,

FIG. 18 is a cross section on the line XVIII-XVIII of FIG. 17,

FIG. 19 illustrates a schematic view in perspective of a rotary screwvolume machine with a coulisse mechanism with planetary motion of themale member, in which Nf=Nm+1,

FIG. 20 shows a cross section of working chambers of a rotary screwvolume machine with additional male and female members being coaxiallydisposed,

FIG. 21 is an exploded view in perspective, explaining the method oftransforming the motion in the rotary screw volume three-dimensionmachine, the principle of forming envelope curvilinear surfaces of themale and female members, and

FIG. 22 illustrates a scheme, explaining the method of transforming themotion in a contra-rotating screw volume machine with planetary motionof the male member, in which Nf=Nm−1.

The rotary screw volume three-dimension machine of FIG. 1 illustrates acircular progressive motion of male member 10, i.e. an axis of the malemember 10 is able to perform only an orbital revolution motion, andswiveling motion of member 10 is absent, whereas a female member 20 isable to rotate on itself.

The circular progressive motion of the male member 10, an axis of whichXm revolves in orbit of E radius about the fixed axis Xf of femalemember 20, is characterized by that a straight line connecting any twopoints of the male member 10 moves parallel to its initial direction.When the male member 10 moves in a circular progressive motion, itsperipheral velocity about its movable axis Xm is equal to zero, i.e. itsswiveling motion is absent.

In the embodied machine of FIG. 1, the male member is formed of athree-arc screw shape outer surface 12 (Nm=3), whereas the female memberhas a two-arc screw shape inner surface 22 (Nf=2). The outer surface ofthe male member 10 defines a male surface 12 and an inner surface of thefemale member 20 defines a female inner surface 22. The male 12 andfemale 22 surfaces are helical surfaces having parallel axes Xm and Xfspaced apart by a length E. The male 12 and female 22 surfaces define atleast one working chamber 11 by evolution of linear contacts A₁, A₂ andA₃, of the male 12 and female 22 surfaces and relative displacement ofthe male 10 and female 20 members.

The nominal profile 14 of the male member 10 having an order of symmetryNm=3 with respect to a center Om located on the male axis Xm isrepresented in a cross section of the rotary screw volumethree-dimension machine given on FIG. 2. In the same way, the nominalprofile 24 of the female member 20 has an order of symmetry Nf=2 withrespect to a female center Of located on said female axis Xf, withNf=Nm−1.

As represented on FIG. 2, the male profile 14 is composed of threeidentical lobes that cover the same angular sector with an angle of apexOm equal to 120°. The same appears with the two lobes of the femaleprofile 24 that are diametrically opposed. The number of such lobesgives the order of symmetry.

The female member 20 is hinged in a stationary main body 30 having amain axis X and is mechanically connected to a one-channel transmissionmeans 31, in a pivot link so as to be able to rotate on itself aboutthis main axis X, which is here mixed with its female axis Xf.

The rotary screw volume machine further comprises a crank like mechanismhaving a crank organ 32 which hinged connects the main body 30 and themale member 10, and presenting an eccentricity equal to E. In fact, thecrank organ 32 is composed by a first shaft like end 32′ hinged in themain body 30 and a second shaft like end 32″ which is parallel, butbrought out of the first shaft like end 32′ with the distance E. Thus,the first shaft like end 32′ is aligned with the axis X which correspondto the driving axis of crank organ 32, and the second shaft like end 32″is aligned with the driven axis of this crank organ 32 which is coaxialwith the axis Xm, while being offset of a distance E with respect to themain axis X.

The male member 10 is hinged on this second crank like end 32″, so asthis second crank like end 32″ is able to revolve about the fixed femaleaxis Xf, i.e. its center Om is able to describe a circle having a radiusE and a center Of.

Consequently, the axis Xm of the male member 10 performs an orbitalrevolution motion about the female axis Xf, which is aligned with themain axis X, whereas the female member 20 rotates on itself about themain axis X of the stationary body 30.

To obtain two dependent degrees of freedom of the male member 10, thecrank organ 32 and the female member 20 are able to be in independentmotion.

When used as an engine, the rotary screw volume machine transforms theenergy coming from the volumetric displacement of a working medium intoa mechanical energy, while when it is used as a pump for example, ittransforms the mechanical energy of means 31 which further comes fromthe motion of the crank organ 32 in the volumetric displacement of aworking medium. To increase the efficiency of such a volume machine,both crank organ 32 and female member 20 can be performing a rotationalmotion.

The screw volume machine further comprises a main synchronizing couplinglink in the form of crank organ 32 and additional mechanism ofsynchronization in the form of crank organ 34 parallel to crank organ 32and gears 36, 38, 40.

The kinematics coupling between the female member 20 and the crank organ32 provides a revolution of the crank organ 32 on rotating female member20 driven by transmission one-channel rotational transmission means 31.

However, because the symmetry order Nf is Nm−1, the synchronization isnot carried out by self-meshing of the elements, it is necessary toprovide a kinematic coupling which can be chosen in the form of reducingor multiplying gear drive.

Consequently, the rotary screw machine comprises a kinematic couplingbetween the female member 20 and the crank organ 32 to allow the motionof the crank organ 32 on rotation of the female member 20. Asrepresented on FIG. 1, the kinematic coupling can comprise at least onecoupling organ 36, such as a toothed wheel, hinged in a pivot link inthe body 30, able to engage on one hand with an internal ring gear 38provided on the female member 20 and on the other hand with a gear 40provided on the crank organ 32.

The trochoidal machine further comprises an additional crank 34 allowingthe circular progressive motion of the male member 10 and the revolutionof the male axis Xm about the female axis Xf.

Each crank 32, 34 comprises a first crank like end 32′, respectively 34′and a second crank like end 32″, respectively 34″. The first crank likeend 32′ cooperates with gear 40, respectively crank like end 34′ withthe body 30, and the second crank like end 32″, respectively 34″, ishinged in the male member 10 and which is parallel, but brought out ofthe first crank like end 32′, 34′ with the distance E. The male member10 cooperates with both crank like end 32″ and 34″, so as male member 10is able to execute circular progressive motion, i.e. its axis Xm is ableto describe a circle having a radius E and a center Of. Theeccentricities E of the crank organ 32 and of the crank organ 34 areequal.

The coupling organ 36, 38 and 40, and the crankshaft 34 form thesynchronizer, which allow the synchronization of the male swiveling andthe female rotation motions.

The transmission ratio between the crank organ 32 and the male member 20is determined by gear wheels 36, 38 and 40 and in particular by thenumber of teeth Z38 and Z40 of gears 38 and 40. The angular cycle isperformed per 180 angular degrees of rotation of member 20, when$\frac{Z\quad 38}{Z\quad 40} = 2.$

When used as an engine, the screw volume machine of FIG. 1 converts theenergy of a working substance into a mechanical energy transmitted tomeans 31. On the opposite, when the machine is used as a pump forexample, it converts the mechanical energy coming from means 31 into aworking substance energy.

FIG. 3 illustrates the version of three-dimension rotary screw volumemachine with the circular progressive motion of the male member 110,which operates similarly to the machine shown in FIG. 1, but with adifferent ratio of number of symmetry between the male and the femalesurfaces. Here, the outer surface 112 of the male member 110 has theform of two-arc trochoid 114 (Nm=2) in a cross-section (see FIG. 4),whereas the inner surface 122 of the female member 120 is in the form ofthree-arc outer envelope 124 (Nf=3) in a cross-section (see FIG. 4).

Here again, the male member 110 is cooperating with the crank organ 32and the crank 34 to perform a circular progressive motion, i.e. the axisXm of the male member 110 is able to perform an orbital revolutionmotion, whereas the female member 120, hinged in pivot link with in thestationary body 30, is able to rotate on itself.

However, in this case, due to the fact that the number of shape-formingarcs is higher for the female 124 (Nm+1), than for the male surface 122,the female 120 and the male 110 members form a kinematic pair, whichprovides self-synchronization.

The volume machine of FIG. 3 operates in the following manner.

When swiveling the crank organ 32 (FIG. 3), due to the cooperation withthe crank 34, the male member 110 executes the circular progressivemotion, the male axis Xm describes a cylinder having a radius E aboutthe female axis Xf, but the male member does not swivel on itself.

As a result of the motion of the male member 110, a self-meshing of themale surface 112 with the inner surface 122 of the female member 120takes place, thus leading to the rotation, in the same direction as thecrank organ 32, of the female member 120 on itself about its axis Xf,which is aligned with the main axis X of the body 30.

FIG. 5 illustrates the version of three-dimension screw volume machinewith a circular progressive motion of the male member 110, and FIG. 6 isa cross section on the line VI-VI of FIG. 5, which operates similarly tothe machine shown in FIG. 3 (Nm=2 and Nf=3), but with a differentconnection of the one-channel rotational means 31 and two parallelcranks 34 instead of only one.

On one hand, here again, the male member 110 is cooperating with atleast two parallel cranks 34 to perform a circular progressive motion.On the other hand, here there is no crank organ 32 and it is the femalemember 120 hinged in pivot link in the stationary body 30, which is ableto rotate, driven by the one-channel transmission means 31. Each crank34 comprises a crank like end 34′ hinged in the body 30 and a crank likeend 34″ hinged in male element 110. The cranks 34 are parallel to oneanother and have the distance E between 34′ and 34″. The male member 110cooperates with the two crank like end 34″, so to be able to execute acircular progressive motion of male element 110, when axis Xm revolvesin a circle having a radius E and a center Of. Here, the eccentricitiesof cranks 34 are chosen to be equal to E.

The female member 120 being directly driven by the one-channel means 31,there is no need of a specific crank organ 32 as describe in FIG. 3. Infact, here the cranks 34 perform as the crank like mechanism.

The rotary volume machine of FIG. 5 operates in the following manner.When means 31 rotates the female element 120 with the angular speed ω₁about its axis Xf, which coincides with the main axis X of the body 30,the inner surface 122 of female member 120 interacts with the outersurface 112 of the male element 110, thus leading to the circularprogressive motion of male element 110 in the same direction as female120 on parallel cranks 34. When the male member 110 executes thecircular progressive motion, the male axis Xm describes a circle havinga radius E and a center Of, with the angular speed ω₀ of a revolution,but the male member 110 is not swiveling (ω₂=0).

In this case, $\frac{\omega_{0}}{\omega_{1}} = 3$and ω₂=0 and an angular cycle measured on rotation (element 120) isequal to 180°.

FIG. 7 represents another version of embodiment of a three-dimensionrotary screw volume machine with two degrees of freedom of which one isindependent. Here as for FIG. 1, the female member 20 is able to performa circular progressive motion, whereas the male member 10 connected to aone-channel rotational means 31 is able to rotate on itself about itsmale axis Xm, which is coaxial with the main axis X.

Here again, because the number of shape-forming arcs of the femaleprofile 24 is lower, than those of the male profile 14 (Nf=2 and Nm=3,see FIG. 8), it is necessary to provide kinematic coupling between themale 12 and the female 22 surfaces.

The male member 10 extends on one end with a shaft 42 on which anexternal ring gear 44 is mechanically fixed. The other end of the malemember 10 is hinged in the main body 30 with a pivot link so as to beable to rotate about the main axis X. The external ring gear 44 iscontinuously meshing with a plurality of gears 46 hinged in the mainbody 30 in a pivot link, so as to drive these gears 46 in rotationalmotion on themselves. The number Z44 and Z46 of teeth of gears 44 and 46is chosen such that $\frac{Z\quad 44}{Z\quad 46} = 3.$Each gear 46 is provided with a crankshaft 48 which is off-center fromthe axis 46′ of each gear 46 of a length equal to E. The parallelcrankshafts 48 are placed in a pivot link in the female member 20.

The elements 42, 44 and 46 have to be compared to the crank organ 32,the gear 30, gears 36 and the internal ring gear 38 of the machine ofFIG. 1.

The operation of the volume machine shown in FIG. 7 proceeds with thecircular progressive motion of the female member 20. In this machine,when the male member 10 is driven by the rotational means 31, it rotatesthe gear wheels 44 and 46 and thus revolves the crankshafts 48. Due tothe rotation of the crankshafts 48, the axis Xf of the female member 20performs an orbital revolution motion about the male axis Xm, i.e. thefemale center Of describes a circle having a radius E and a center Om inthe same direction as the male member 10.

In the versions of the machine embodiments aforementioned, the choice ofthe eccentricity E has no effect on the values of diameters of thesynchronizing gear wheels 36, 38, 40 of FIG. 1 and 44, 46 of FIG. 7.

FIG. 9 illustrates a rotary screw volume machine similar to the rotaryscrew machine of FIG. 1, but with three degrees of freedom, two of thembeing independent. This rotary screw volume machine comprises the femalemember 20 of screw shape (two arcs), the three-arcs male member 10 (seeFIG. 10), the stationary body 30, the crank like mechanism comprisingthe crank organ 32 hinged with a pivot link in the main body 30 havingthe main axis X, so that the axis Xm of the male member 10 is able torevolve about the female axis Xf which is aligned with the main axis Xand the female member 20 is able to rotate with rotational means 131about the main axis X.

Because the symmetry order Nf is Nm−1, the synchronization is notcarried out by self-meshing of the elements, it is necessary to providea kinematics coupling between the male and the female members.

Consequently, the crank organ 32 and the female member 20 can be linkedto a two-channel rotational transmission means 131. The female member 20is connected to one of the two channels of the rotational transmissionmeans, whereas, the crank organ 32 is connected to the other one of thetwo channels of the rotational transmission means.

Under two-channel connections of means with two independent degrees offreedom of the machine, any two angular rotation velocities of thefemale member 20 or the crank organ 32 can be specified (independentdegrees of freedom), whereas the third swiveling angular rotationvelocity of the male member 10 (dependent degree of freedom) is set inthe machine as a differential function of the two independentvelocities. In this case, additional synchronizing means are not needed.

On the opposite, under one-channel transmission means 31 (see FIG. 11),the coupling with a machine would be performed through one channel ofindependent degree of freedom, and an additional synchronizing meansshould be introduced into the machine to connect any two of the threemachine elements (male member 10, female member 20 or crank organ 32)with the feasibility to decrease the quantity of independent degrees offreedom of machine by unity.

The additional degree of freedom is the swiveling motion of the femalemember 20.

For example, as represented on FIG. 9, the male member 10 provides atone end an internal ring gear 50 that engages with a pinion 52 rigidlyfixed on the female member 20 and hinged in the main body 30 so as to beable to rotate with means 131. The planetary gear transmission 50 and 52connects respectively mechanically the male member 10 and the femalemember 20, whereas both crank organ 32 and female member 20 areconnected to a two-channel rotational means 131.

Due to the different gears, when the crank organ 32 rotates in adirection, the male member 10 performs an orbital revolution in asimilar direction, i.e. the male axis Xm describes a circle of center Ofin the same direction of rotation as the crank organ 32, whereas themale member 10 swivels on itself in the opposite direction of rotation.In fact, the orbital revolution of the male axis Xm and the swivelingmotions of the male member 10 are in opposite direction.

To obtain a contra-rotating rotary screw three-dimension volume machine,i.e. the revolution speed of the female member 20 and the orbitalrevolution speeds of the crank 32 and the male axis Xm are equal, but inan opposite direction, the different gears can for example be chosen asfollows. The internal ring gear 50 has an internal radius equal to threetimes E, 3×E, the outer gear 52 has an external diameter equal to 2×E.Thus, the ratio of the number of teeth Z50 and Z52 of each gear 50 and52, is chosen so that $\frac{Z\quad 50}{Z\quad 52} = {\frac{3}{2}.}$

The operation of the contra-rotating rotary screw three-dimension volumemachine of FIG. 9 proceeds as follows. With help of rotational means131, when rotating the crank organ 32 and simultaneous female member 20,on one hand, due to the crank organ 32, the male member axis Xm performsthe orbital revolution motion about the main axis X, and on the otherhand, due to the interaction of internal ring gear 50 of the male member10 with external gear 52 connected to the female member 20, the malemember 10 execute the swiveling motion on itself. The combination ofboth motions, swiveling and orbital revolution of the male axis Xm,springs up the planetary motion of the male member 10.

The efficiency of the screw machine being proportional to the speed ofthe processes of opening and closing the chambers between the conjugatedsurfaces of male and female members is determined by the duration of theangular cycle of the machine. In this machine represented on FIG. 9, theangular cycle is equal 270 angular degrees, that is twice as less thanin the known machines of this type, because it is performed, when twomembers forming the working chambers are in a relative simultaneousmotion.

However, the best result for the machine of FIG. 9 is obtained when therevolution speed of an axis of member 10 is equal to the rotation speedof member 20 and occurs in the opposite direction of rotation. In thiscase, the mechanical strengths produced by rotating female 20 and by arevolution of crank 32 with male member 10 on the main body 30 are equaland opposite, such that the resultant momentum is practically nil. Thesekinds of machines are used in cases where the vibrations are to beavoided or greatly limited.

FIG. 11 illustrates a rotary screw volume machine similar to the rotaryscrew machine of FIG. 9, but with three degrees of freedom, one of thembeing independent and with one-channel rotational means 31. This rotaryscrew volume machine comprises the female member 20 of screw shape (twoarcs), the three-arcs male member 10 (see FIG. 12), the stationary body30, the crank like mechanism comprising the crank organ 32 hinged with apivot link in the main body 30 having the main axis X, so that the axisXm of the male member 10 is able to revolve about the female axis Xfwhich is aligned with the main axis X and the female member 20 is ableto rotate on itself about the main axis X.

To avoid having the rotational means connected to both crank organ 32and female member 20 and because the number of shape-forming arcs of thefemale profile 24 is lower than those of the male profile 22, the rotaryscrew machine comprises a planetary gear transmission. According thedisposition of both gears internal/external engagement, the planetarygear transmission 50, 52, drives the female member 20 in the samedirection or in the opposite direction relative to the crank organmotion.

To provide this additional motion, the rotary screw machine comprises anadditional synchronizer, which comprises a planetary gear transmission.It is also possible to make the additional synchronizer in the form of acoulisse mechanism with a rotating or fixed coulisse or an inverter of amotion direction.

For example, as represented on FIG. 11, the male member 10 provides atone end an internal ring gear 50 that engages with a pinion 52 rigidlyfixed on the female member 20 and hinged in the main body 30.

To synchronize the different motions between the male 10 and female 20members, the rotary screw machine further comprises a synchronizer. Forexample, the male member 10 provides at its other end a pinion 54, whichengages with an internal ring gear 56, fixed in the main body 30.

Due to the different gears, when the crank organ 32 rotates in adirection, the axis Xm of the male member 10 rotates in a similardirection, i.e. the male axis Xm describes a circle of center Of in thesame direction of rotation as the crank organ 32, whereas the malemember 10 swivels on itself in the opposite direction of rotation. Infact, the orbital revolution of male axis Xm and the swiveling motionsof the male member 10 are in opposite direction.

To obtain a contra-rotating screw three-dimension volume machine, i.e.the rotational speed of the female member 20 and the orbital revolutionspeed of the male axis Xm are equal but in an opposite direction, thedifferent gears can for example be chosen as follows. The internal ringgear 50 has an internal radius equal to three times E, 3×E, the outergear 52 has an external radius equal to 2×E. Thus, the ratio of thenumber of teeth Z50 and Z52 of each gear 50 and 52, is chosen so that$\frac{Z\quad 50}{Z\quad 52} = {\frac{3}{2}.}$The internal ring gear 56 has an internal radius equal to 4×E, the outergear 54 of the male member 10 has an external radius equal to 3×E.

Thus, the ratio of the number of teeth Z₅₆ and Z₅₄ of each gear 56 and54 is chosen so that $\frac{Z\quad 56}{Z\quad 54} = {\frac{4}{3}.}$

The operation of the contra-rotating screw three-dimension volumemachine proceeds as follows. When rotating the crank organ 32 (via theone-channel rotational means 31), on one hand, the axis Xm of the malemember performs the orbital revolution motion about the main axis X, andon the other hand, the gear 54 of the male member 10 is rolled on theinner surface of the stationary internal ring gear 56 and thus makes themale member 10 execute the swiveling motion on itself. The combinationof both motions, swiveling and orbital revolution, springs up theplanetary motion of the male member 10. Moreover, the internal ring gear50 rotates the gear 52 of the female member 20, which rotatescontra-rotatively according to the crank organ's direction.

FIG. 13 shows a longitudinal section of a contra-rotating screw volumemachine with one independent degree of rotation of the female member 20,in which Nf=Nm−1, and FIG. 14 is a cross section on the line XIV-XIV ofFIG. 13, similar to the screw machine of FIG. 11 (Nf=2 and Nm=3), butwith a different connection of the one-channel rotational means 31.

The male member 10 is able to execute a planetary motion about thefemale axis Xf, which coincides with the main axis X and the femalemember 20 is able to rotate about the main axis X and connectedmechanically to one-channel transmission means 31.

The female member 20 has a profile 24 and male member 10 has a profile14. The screw machine comprises the same planetary gear transmissions54, 56 as described in FIG. 11, but another planetary gear 150, 152replace the former planetary gear 50, 52 aforementioned.

According the disposition of both gears internal/external conjugation,the planetary gear transmission 150, 152 has the relation${\frac{z_{150}}{z_{152}} = \frac{3}{2}},$where z₁₅₀ and z₁₅₂ represent respectively the number of teeth of gears150, 152. Accordingly, herewith, gear 152 (outer conjugation) isdisposed on female member 20 and connected to the one-channel means 31and gear 150 (inner conjugation) is disposed on male member 10.

The independent degree of freedom is the rotation of the female member20, and the dependent degrees are the motion of male member 10(swiveling of its member and revolution of its axis Xm). To create thesetwo dependent motions, the machine comprises the additional synchronizercomprising the planetary gear transmission 54, 56 aforementioned. Forexample, the planetary gear transmission 54, 56 has the relation${\frac{z_{56}}{z_{54}} = \frac{4}{3}},$where Z₅₆ and Z₅₄ represent respectively the number of teeth of gears56, 54.

Due to said gears, the axis Xm of male member 10 performs a revolutionin opposite direction of the swiveling of the male member 10 about itsmale axis Xm and describes a circle having a radius E and a center Of.The female member 20 executes a rotation about fixed axis Xf in oppositedirection of the revolution of the male axis Xm.

The speed of the female member 20 and the rotation speed of the maleaxis Xm are equal, but have opposite direction. The different gears canfor example be chosen as follows. The internal ring gear 150 has aninternal radius equal to 3×E (three times E), the outer gear 152 has anexternal radius equal to 2×E. The internal ring gear 56 has an internalradius equal to 4×E, the outer gear 54 of the male member 10 has anexternal radius equal to 3×E.

The operation of the screw three-dimension volume machine proceeds asfollows. When the female member 20 and the gear 152 rotate, due to theirconnection to the one-channel rotational means 31, the male member 10and the gears 150 and 54 execute a planetary motion about the main axisXf. As the gear 54 of the male member 10 is rolled on the inner surfaceof the stationary internal ring gear 56, the male member 10 execute aswiveling about its axis Xm and its axis Xm executes a revolution aboutaxis X. Moreover, the internal ring gear 152 rotates the gear 150 of themale member 10, creating a revolution of its axis Xm at an angularvelocity equal to velocity of female element 20, but in oppositedirection.

The angular cycle of the machine described on this FIG. 13 is equal 270°of an angular turn of the female element 20.

FIG. 15 shows a longitudinal section of another version of embodiment ofa rotary screw of three-dimension volume contra-rotating machine withthree degrees of freedom and two-channel rotational means 131. In fact,this machine has to be compared to the abovementioned machine (FIG. 9)in which the male member 110 is performing a planetary motion and thefemale member 120 is rotating on itself, but now the male member 110 hasa nominal profile 114 composed of two arcs and the female member 120 hasa nominal profile 124 composed of three arcs (see FIG. 16).

In this case, due to the fact that the number of shape-forming arcs ishigher for the female profile 124 (Nf=Nm+1), than for the male profile114, the female 120 and the male 110 members form a kinematics pairwhich provides self-synchronization and synchronizing coupling betweenthe female 120 and the male 110 members, such as the kinematics couplingof gear wheels 50 and 52 of FIG. 9, is not needed.

Two outlets of the two-channel transmission means 131 are respectivelyand mechanically connected to female member 120 and crank 32 to create arotation (first independent velocity) of female member 20 about itsfixed axis Xf and a revolution (second independent velocity) of maleaxis Xm about the main axis X so as to define a contra-rotating machinehaving a resultant momentum almost nil.

This machine operates similarly to the machine shown in FIG. 9. The malemember 110 is hinged on crank 32 and performs a swiveling about its axisXm when the crank organ 32 rotates, and the female member 120 hinged inbody 30 is able to rotate about the main axis X.

The two-channel rotational means 131 creates the two independentvelocities of a rotation for female member 120 and a revolution forcrank organ 32, which are equal to one another but have oppositedirection.

Thus, when crank 32 revolves, the male member 110 executes a planetarymotion in the process of which due to the self-synchronization maleprofile 114 interacts with the female profile 124, then male member 110swivels (third dependent velocity) about movable axis Xm. The malemember 110 swivels in the same direction as the female member 120. Theangular cycle of the machine of FIG. 15 is equal 180 degrees of anangular turn of the female member 120 or the crank organ 32.

In the machines described on FIGS. 9 and 15, there are three degrees offreedom of which the two ones are independent and the transmission ofpositive energy of conversion is performed by the two-channel means 131through two mechanical channels of independent rotation or revolution.

Any two angular speeds of motions of said three ones (rotation,revolution or swiveling of male or female member, or synchronizingcoupling link) can be specified as independent of one another. Theinitial phase and direction of each rotation are defined, and the valuesof said angular speeds are chosen in conformity with the equations:k ₁ω₁ +k ₂ω₂+ω₃=0,where:

ω₁,ω₂ represent the angular speed of the said conjugated members abouttheir axis;

ω₃ represents the angular speed of the link of synchronizing coupling;

k₁,k₂ represents the constant coupling coefficients. Herewith, thevalues of angular velocities of rotation of conjugated members aredefined from relation:(z−1)ω₁ −zω ₂+ω₀=0,where:ω₁ is the angular speed of member around its axis, enveloping surface ofwhich has the form of curvilinear surface;ω₂ is the angular speed of rotation of member around its axis,enveloping surface of which has a shape of inner or outer envelope of afamily of surfaces, formed with the said curvilinear surface;ω₀ is the angular speed of the orbital revolution of the axis of themember, executing planetary motion;z is an integer, z>1.

FIG. 17 shows a longitudinal section of another version of embodiment ofa rotary screw of three-dimension volume contra-rotating machine withthree degrees of freedom and one-channel rotational means 31. In fact,this machine has to be compared to the abovementioned machine of FIG. 11in which the male member 10 executes a planetary motion and the femalemember 20 rotates on itself, but now the male member 110 has a nominalprofile 114 composed of two arcs and the female member 120 has a nominalprofile 124 composed of three arcs (see FIG. 18).

An inverter 58 can be placed between the female member 120 and the crankorgan 32 to invert the motion direction between the rotational motion ofthe female member 20 on itself and the orbital revolution motion of themale axis Xm about the main axis X so as to define a contra-rotatingmachine having a resultant momentum almost nil.

This machine operates similarly to the machine shown in FIG. 11. Themale member 110 cooperates with the crank organ 32 and performs aplanetary motion about the main axis X, and the female member 120 ishinged in the body 30 and is able to rotate on itself about the mainaxis X. The female member 120, through the direction motion inverter 58is mechanically connected with the crank organ 32. The inverter 58 leadsto the same speed for the female member 120 and for the crank organ 32,i.e. for the orbital revolution of the male axis Xm, but the two motionsoccur in opposite direction.

When rotating the crank organ 32 (via the one-channel rotational means31), the male member 110 executes the planetary motion; due to theself-synchronization taking place when the male profile 114 interactswith the female profile 124, the female member swivels on itself. Therotation of crank organ 32 through the inverter 58 causes the rotationof the female member 120 at the same angular speed as the rotation speedof this crank organ 32, but in the opposite direction. The male member110 swivels in the same direction as the female member 120 rotates.

FIG. 19 illustrates the version of a three-dimension rotary screw volumemachine with a planetary motion of the male member 110, which operatessimilarly to the machine shown in FIG. 9, but with a different ratio ofvelocities. In FIG. 19, there is one independent degree of freedom, i.e.the rotation of the female member 120. The swiveling and the revolutionof male member 110 are dependent motions. The angular speed of aswiveling of male member 110 is equal to −3 arbitrary units, and theangular speed of a revolution of its axis Xm is equal to +3 arbitraryunits, i.e. they are equal in values but opposite in direction. Theangular speed of rotation of female member 120 about its fixed axis Xfis equal to −1 arbitrary units. Here, the outer surface 112 of the malemember 110 has the form of two-arc trochoid (Nm=2) in a cross-section,whereas the inner surface 122 of the female member 120 is in the form ofthree-arc outer envelope (Nf=Nm+1=3).

Here again, the male member 110 is mechanically rigidly connected to acrank organ 59, the main crank 59″ of which is mechanically rigidlyconnected to male member 110 in a point 62. The point 62 has thecoordinates (0; E), when the male center Om is taken as an initialposition of coordinate system. A crankpin 59′ of the crank organ 59extend at 2E distance from the main crank 59″ and is disposed along thefemale axis Xf.

Two sliders 60 are hinged on the main crank 59″ and on the crankpin 59′,with the possibility to slide in rectilinear grooves, e.g. in twocoulisses 61 provided in the fixed body 30. The longitudinal axes ofthese coulisses 61 are perpendicular.

Taken in combination, the crank organ 59, the sliders 60 and thecoulisses 61, form an ultimate coulisse mechanism intended for creatinga planetary motion of the crank organ 59 together with the male member110 relative to the body 30 about the female fixed axis Xf. The femalemember 120 is hinged in the body 30 and is mechanically connected to aone-channel transmission means 31 and is able thus to rotate by thismeans about its fixed axis Xf.

However, in this case, due to the fact that the number of shape-formingarcs is higher for the female 122, than for the male surface 112(Nf=Nm+1), the female member 120 and the male member 110 form akinematics pair with self-synchronization only with availability of thecoulisse mechanism 59, 60, 61 providing a planetary motion of malemember 110.

The rotary volume screw machine of FIG. 19 operates in the followingmanner. When the one-channel rotational means 31 rotates the femalemember 120 about its fixed axis Xf, then due to the cooperation ofcurvilinear surfaces 122 and 112, and cooperation of the crank organ 59,the sliders 60 and the coulisses 61, the male member 110 executes theplanetary motion, i.e. the male axis Xm revolves in a circle having aradius E and a center Of, the sliders 60 execute a reciprocating motionwith an amplitude 4E in the coulisses 61. As a result of the swivelingand revolution of the male member 110 with the same velocities, aself-meshing of the male surface 112 with the inner surface 122 of thefemale member 120 takes place, thus leading to the same direction ofswiveling of the male member 110 about its movable axis Xm and rotationof female member 120 about its fixed axis Xf, which coincides with themain axis X of the body 30.

An angular cycle of machine of FIG. 19 is equal to 90 angular degrees ofturn of female member 120.

To increase the efficiency of such kind of three-dimension rotary screwvolume machine, it is also possible to increase the number of male andfemale members, which can be coupled to one another mechanically orthrough the working medium. The additional male and female members canbe disposed in line with said male and female members or can be disposedcoaxially inside said male and female members as illustrated in FIG. 20,in such a way that their surfaces are in mechanical contact so as toform additional chambers.

Referring to the FIG. 20, in which for example, four members 500, 600,700 and 800 engage in each other. A first two-arc member 500 (male) isengaging in the inner three-arc profile 624 (outer envelope of a family)of a first three-arc member 600. This first three-arc member 600 is afemale member for the first two-arc member 500, but is a male member forthe second two-arc member 700 in the inner profile 724 of which theouter profile 614 (inner envelope of a family) of this first femalemember 600 is engaging. It occurs the same with this second two-arcmember 700, which is also male and female, and which outer profile's 714(two-arc initial trochoid) is engaging in the inner three-arc profile824 (outer envelope of a family) of a last three-arc member 800. In thisparticularly case, the member 700 can be mechanically connected to themember 500, and the member 600 to the member 800, and the number ofworking chambers 11, has increased from three to nine.

The three-dimension rotary screw volume machine can comprises at leastone additional male and female members disposed in line (notillustrated) and mechanically rigidly connected to said main male andfemale members herewith forming additional working chambers.

Moreover, all the three-dimension rotary screw volume machines abovedescribed can have male and female surfaces degenerated into cylindricalsurfaces.

We will now explain how the medium is displacing in the working chambersof such a three-dimension rotary screw volume machine.

The interconnected rotary motion of a link of synchronizing couplingand, at least, two sets of enclosing and being enclosed conjugatedelements is executed. In the initial state, the elements of sets turnabout their common fixed axis relative to each other; with thefeasibility to form set of volumes between the male and female members,that jointly form the total working chambers. These volumes are limitedby the surfaces made in the shape of cycloid or trochoid, or in theshape of fragments of said surfaces, which taken jointly form the totalworking (displacing) chambers.

Two motions of said three ones (swiveling and orbital revolution of themale member, and rotation of the female member) are independent of oneanother.

For example, referring to FIG. 21, seven elements 10 n fixed together soas to form the three arcs male member 10 of FIG. 11 with vertices A₁,A₂, A₃, and the male profile 12 is made in the form of the outer surface(Nm=3). Seven elements 20 n form also together the female member 20,which defines the inner surface. Each element of female member 20 has across section, which is limited radially by a cylindrical surface havingan order of symmetry Nf about the female axis Xf (e.g. in the shape oftwo-arc epitrochoid, Nf=Nm−1=2). The number of intersecting points ofthe inner and outer surfaces z is equal to three (z=3). The axes Xm andXf are spaced apart by a distance E (eccentricity).

FIG. 21 illustrates also, in a diagram, the seven angular positions a,b, c, d, e, f and g of the seven elements composing each member male 10or female 20 according to the length L of the machine. The male andfemale elements are turned around their axis, respectively Xm and Xf, inone direction. The period Pm represented by b-f, on which the totalworking chamber is made, i.e. at mentioned section a period of totalvariation of an area of the end section of the working chamber isperformed, i.e. it corresponds to a complete opening and closure of aworking chamber.

The ratio of periods of birotative turn of male and female elements ofconjugated sets is equal to Nm/Nf=3/2. The male and female elements formthe three total working chambers and define three areas S_(A1A2),S_(A2A3), S_(A3A1) of end sections of which vary with a spatial shiftPm/3.

The ratio of turn angles of the elements on the period b-f of turn, orthe axial period of total volumes, is chosen proportionally with theratio of the orders of symmetry of shapeforming arcs of the profiles 14and 24, so that at z turns of female member 20 (trochoid), there wouldbe z−1 turns of the male member 10 (internal envelope), with feasibilityto form the total displacing working chambers with the closed areasS_(A1A2), S_(A2A3), S_(A3A1) taken in a cross section.

In position b, taken as an initial position, closed area S_(A2A3) has aminimal value. In position c, the elements 10 n of the male member 10,are turned about their male axis Xm in clockwise direction through anangle (φ_(m)=90°, and the elements 20 n of the female member 20 areturned around Xf axis through an angle of φ_(f)=135°. The ratio of turnangles φ_(f)/φ_(m) is equal to 3/2.

In position d the turn angles, relative to initial position b are equal180° for the male member 10 and 270° for the female member 20, etc. Forexample, the closed area S_(A2A3) has a maximal value in position d.

When the male member 10 and the female member 20 execute the aforesaidturns, all elements of male and female members taken in combination ateach turn and in relation with their specific thickness and positionside by side, form the total working chambers with a discreet stepthree-dimensional change of the volumes and with the feasibility ofaxial motion of the volumes of working chambers.

In increasing the number of elements up to infinity and decreasing theiraxial thickness up to zero defining curvilinear conjugated surfaces, thethree-dimension changes along the axis of the volumes of total workingchambers between the male 10 and the female 20 appear smoothly.

According to the number of elements, the number of arcs and the speedand direction of rotation motion, the axial period of total volumes willdiffer.

The conjugated pair of male 10 n and female 20 n elements isself-sufficient. The process of an axial motion from chamber to chamber,carries out different thermodynamic transformations (compression,expansion and so on) of different working media, that is why the processof axial motion of the volumes from one working chamber 11 to anotherone can be done without using end walls, additional bodies, elements forgas distribution, valves, etc.

In FIG. 21, there are three of such volumes and the spatial phase shiftbetween them is equal to 120°. The scheme of FIG. 22, explains themethod of transforming the motion in rotary screw volume machine inwhich the male member 10 is in planetary motion in a female member 20,which is rotating about the main axis of the machine.

The male member 10 having an Nm order of symmetry revolves, i.e. itsaxis Xm describes a portion cylinder having a radius equal to E and atan angular speed ω₀=+ω through an angle θ about the female axis Xf.Moreover, at fixed female member 20, the male member 10 swivels onitself at an angular speed +ω/3 about its axis Xm in the same directionas its orbital revolution motion, so that the three vertices A₁, A₂ andA₃ slide on the epitrochoid profile 24 of the female member 20 incontinuous contact with it. The inner surface of the female member 20 islimited radially by a cylindrical surface having an order of symmetryNm−1 (e.g. two-arc epitrochoid).

In a planetary motion of the male member 10, whereas the female member20 is stationary, the working volumes considered in a cross sectiondescribe a circle and the total working volumes execute axial motionalong the longitudinal axes of the elements.

In the initial position, the male member 10 has a period b-f (Pm) of ascrew turn about the male axis Xm, and the female member 20 has a periodPm=3/2 Pm about axis Xf. In FIG. 21, the period b-f is equal to a periodof a complete opening and closure of a working chamber. When the femalemember 20 is fixed, an angular speed of a revolution of the male memberaxis Xm is equal to ω₀=ω, and the angular speed of a swiveling of themale member 10 about its movable axis Xm is equal to$\omega_{2} = {\frac{\omega_{0}}{3} = {\frac{\omega}{3}.}}$

According to the invention, as the independent motions any two of thethree motions of male and female members and synchronizing coupling linkcan be determined, we determine a counter-rotative revolution of axis Xmof the male member 10 (carried out by crank mechanism which is not shownin FIG. 21) at ω₀=+ω and additional rotation of the female member 20about fixed axis Xf at ω₁=−ω, i.e. revolution of the crank mechanismabout axis Xf and an axis Xm of the male member 10 at +ω is performedsimultaneously.

The dependent angular speed ω₂ is swiveling of the male member 10 aboutmovable axis Xm and is determined by the equation mentioned above (atz=3): (3−1)(−ω)−3ω₂+ω=0. Whence $\omega_{2} = {- {\frac{\omega}{3}.}}$

An angular cycle of the axial movement of one closed volume between themale and female members in the planetary method of transforming a motionat fixed female member 20 is performed per 540° of a revolution of maleaxis Xm about the axis Xf of the female member 20.

According to the invention an angular cycle measured on rotation(element 20) or on revolution (crank) is θ=270°, and the angular cyclemeasured on swiveling (element 10) is$\Psi = {\frac{\theta}{Nm} = {90{{^\circ}.}}}$

We have seen that the additional independent degree of freedom ofrotational motion of the female elements is brought when three rotarymotions are made, two of them are independently chosen. The initialphase and direction of each rotation are defined, and the values ofrotation angular speeds of said sets of conjugated elements are chosenin conformity with the equations: $\{ \begin{matrix}{{{K_{1}\omega_{1}} + {K_{2}\omega_{2}} + \omega_{3}} = 0} \\{{{( {z - 1} )\omega_{1}} - {z\quad\omega_{2}} + \omega_{0}} = 0}\end{matrix}\quad $where

ω₁, ω₂ are the rotational speeds of said male and female members onthemselves about their axis;

ω₃ is the rotational speed of the synchronizing coupling link;

K₁, K₂ are constant coupling coefficients,

ω₀ is the angular speed of revolution motion of the male axis Xmrotating about the female axis Xf;

z is the number of cross points A₁, A₂, A₃, etc. of inner and outerenvelopes of said male and female surfaces, and can be any integer whichis more than unity.

Any two of the angular independent speeds can be chosen in an arbitraryway, coefficients and the third dependent speed are determined by theequations given above.

After specifying the values of the two independent speeds and z value,they should be substituted into the equations mentioned above, so as toobtain the values of the dependent speed and the constant coefficients.

To create an additional independent degree of freedom of rotary motionof the conjugated elements an additionally birotative motion of bothmembers is introduced. As shown in FIG. 22, the male member 10 and thefemale member 20 rotate additionally about their centers Om and Of inone direction (opposite to a revolution of an axis of the male member)with the angular speeds −⅔ω for the male member 10 and ω₁=−ω for thefemale member 20.

In this case, the male member 10 acquires the overall speed of its ownperipheral swiveling about its center Om, which is equal to$\omega_{2} = {{\frac{\omega}{3} - {\frac{2}{3}\omega}} = {{{- \frac{\omega}{3}}\quad{and}\quad{the}\quad{angle}\quad{of}\quad{turn}\quad\Psi} = {- \frac{\theta}{Nm}}}}$about Of (an angle Ψ in FIG. 22 denotes a peripheral turn or swivelingabout an axis Xm crossing the male center Om, and angle θ denotes a turnangle of the female member 20 about fixed axis Xf crossing the femalecenter Of). The center of male element Om retains its orbital motionspeed in a circle ω₀=+ω and an angle θ, and the female member 20 isimparted the speed ω₂=−ω. This indicates that in this case the verticesA₁, A₂, A₃ of the three-angular male member will describe a hypotrochoidand at the same time will slide along a female member epitrochoid whichrotates about its center Of with an angular speed −ω.

Other versions of transforming a motion with other combinations ofrotary, planetary and circular progressive motions are possible. Forcontra-rotary variant, we determine ω₀=+1, ω₁=−1, and male member withz=3 inner envelope. Consequently, the substitution of these values inthe equations mentioned, gives k=−1, ω₂=−⅓.

As it is shown in FIG. 22, an angular cycle decreases to −270° of a turnangle of the female member about its axis Xf. It points to the fact thatthe angular duration of the cycle decreases by an half in comparisonwith the known closest analogue of the planetary method of transforminga motion with the stationary epitrochoid of the female member and themale member with three vertices, thus the number of cycles performed pergiven number of revolutions increases two times, this gives rise tointensification of the thermodynamic cycles of the volume machines aswell.

Furthermore, an axis of male member 10 and the female member 20, as itis shown in FIG. 22, rotating in the opposite directions with the equalangular speeds, i.e. counter-rotatively, provide decreasing considerably(up to zero) the combined moment of momentum and reaction moment on thesupports of the machine.

The planetary motion of male member 10 can be described by theexpression:${{\overset{\_}{e}}_{RV} + {\frac{1}{z}{\overset{\_}{e}}_{S}}},$where {overscore (e)}_(RV) and {overscore (e)}S are unit vectors of therevolution and swiveling speeds of male element.

The birotation of the male and female elements is described by thefollowing expression:${k\quad{\overset{\_}{e}}_{R0}} + {\frac{k( {z - 1} )}{z}{\overset{\_}{e}}_{S}}$where {overscore (e)}_(R0) is a unit vector of the rotation angularspeed rotation of the female element 20.

By adding the birotative motion and the planetary motion, we obtain:${k\quad{\overset{\_}{e}}_{R0}} + {\frac{\lbrack {{k( {z - 1} )} + 1} \rbrack}{z}\quad{\overset{\_}{e}}_{S}} + {{\overset{\_}{e}}_{RV}.}$

From the preceding equations, it follows that on executing the profileof the end sections of the member executing the planetary motion in theform of the inner or the outer envelope of a family of curves and theprofile of the member rotating about its fixed axis in the form of theinitial curve, the relation of the angular speed of rotation of thelatter one to the angular speed of a revolution of an axis of theelement executing the planetary motion is equal to k, and the relationof the angular speed of the swiveling motion of the planetary member tothe angular speed of a revolution of its axis is equal to$\frac{\lbrack {{k( {z - 1} )} + 1} \rbrack}{z}.$So, as an example, with z=3, the planetary motion of the male memberwith an inner envelope and an additional rotation of epitrochoid of thefemale member and the male member around their axis, we obtain:1) θ=45°, k=−5, k₁=−5 and k₂=−3 and an angular cycle equal to γ=90° of arevolution of the male member axis about the female center Of.2) θ=135°, k=−1, k₁=−1 and k₂=−⅓ and an angular cycle equal to γ=90° ofa swiveling of the male member about its male center Om.

The following versions of transforming a motion in this mechanism arepossible:

1) without transmission of motion between the female and the malemembers; in this case, their motions are defined by the links ofsynchronization without kinematics interaction of conjugated elements;

2) with the transmission of rotation by interacting conjugated members;in this case, the curvilinear surfaces of female and male members arebrought in mechanical contact, forming a kinematics pair and performingwith said pair the transmission of motion between female and malemembers.

A kinematics conjugation of any number of the additional female and malemembers is possible, which are fitted in the additional means ofsynchronization with the feasibility of the rotary and planetarymotions, herewith the main and additional elements can be placedalongside each other or in the cavities of each other.

1. A rotary screw machine of volume type comprising a body (30) having amain axis X, two members consisting of a male member (10; 110; 500; 600;700) and a female member (20; 120; 600; 700; 800) surrounding said malemember, wherein an outer surface of the male member (10; 110; 500; 600;700) defines a male surface (12; 112) and a inner surface of the femalemember defines a female surface (22; 122), said male (12; 112) andfemale (22; 122) surfaces being helical surfaces having respective axesXm and Xf that are parallel and spaced apart by a length E, said male(12; 112) and female (22; 122) surfaces defining at least one workingchamber (11) by formation of linear contacts (A1, A2, A3) of said male(12; 112) and female (22; 122) surfaces and relative displacement ofsaid male (10; 110; 500; 600; 700) and female (20; 120; 600; 700; 800)members, said male (12; 112) and (22; 122) female surfaces being furtherdefined about said axes Xm and Xf by a nominal profile in a crosssection of the mechanism, said profile of the male surface (12; 112)defining a male profile (14; 114; 514; 614; 714) having an order ofsymmetry Nm with respect to a center Om located on said male axis Xm,said profile of the female surface (22; 122) defining a female profile(24; 124; 624; 724; 824) having an order of symmetry Nf with respect toa center Of located on said female axis Xf, said rotary screw machinefurther having a main synchronizing coupling comprising a crank likemechanism (32; 34; 48; 59) generating an eccentricity E between saidmain axis X and one of the axes (Xm, Xf), characterized in that a firstone of said male (10; 110; 500; 600; 700) and female (20; 120; 600; 700;800) members is hinged in said body (30) and is able to rotate on itselfabout its fixed axis (Xm; Xf) according to a rotational motion, in thatsaid crank like mechanism (32; 34; 48; 59) is connected to a second oneof said male (10; 110; 500; 600; 700) and female (20; 120; 600; 700;800) members to allow the axis (Xf; Xm) of said second member to revolveabout the fixed axis of said first member (Xm; Xf) according to anorbital revolution motion having said length E as a radius, and in thatsaid rotary screw machine comprises a main synchronizer (34, 40, 36, 38;44, 46, 48; 54, 56; 58;) synchronising said swiveling motion and saidorbital revolution motion, one with respect to the other, so that saidmale (12; 112) and female (22; 122) surfaces mesh together.
 2. A rotaryscrew machine according to claim 1, characterized in that it furthercomprises rotational transmission means (31; 131) connected to saidcrank organ (32; 59) or to said first member (10; 110; 500; 600; 700;20; 120; 600; 700; 800).
 3. A rotary screw machine according to claim 2,characterized in that said rotational transmission means (131) is atwo-channel rotational means (131).
 4. A rotary screw machine accordingto claim 1, characterized in that said male (12; 112) and female (22;122) surfaces are brought in mechanical contact forming a kinematic pairallowing the transmission of motion between said first (10; 110; 500;600; 700) and second (20; 120; 600; 700; 800) members.
 5. A rotary screwmachine according to claim 1, characterized in that it further comprisesan additional synchronizer (50, 52), linked to said body and allowingsaid second member (20; 120; 600; 700; 800; 10; 110; 500; 600; 700) torotate about its axis.
 6. A rotary screw machine according to claim 5,characterized in that said additional synchronizer comprises a planetarygear transmission (50, 52).
 7. A rotary screw machine according to claim5, characterized in that it further comprises rotational transmissionmeans (3 1; 13 1) connected to said crank organ (32;34; 48; 59) and toone of said male (10; 110; 500; 600; 700) or female (20; 120; 600; 700;800) member.
 8. A rotary screw machine according to anyone of the claim1, characterized in that said synchroniser further comprises akinematical coupling mechanism (40, 36, 38; 44, 46, 48) of both members(10; 500; 600; 700; 20; 600; 700; 800) together, said kinematicalcoupling comprising at least one coupling organ (36; 46), which ishinged in said body (30).
 9. A rotary screw machine according to claim8, characterized in that said kinematical coupling mechanism comprises agear transmission (40, 36, 38; 44, 46, 48).
 10. A rotary screw machineaccording to claim 1, characterized in that said synchronizer comprisesa planetary gear transmission (54, 56).
 11. A rotary screw machineaccording to claim 1, characterized in that said synchronizer comprisesan inverter (58).
 12. A rotary screw machine according to claim 1,characterized in that said synchronizer comprises a coulisse mechanism(59, 60, 61).
 13. A rotary screw machine according to claim 1,characterized in that it further comprises at least one additional maleand female members (500; 600; 700; 600; 700; 800) disposed in line withsaid male and female members.
 14. A rotary screw machine according toclaim 1, characterized in that it further comprises at least a thirdmember disposed inside or surrounding said male and female members (500;600; 700; 600; 700; 800), in such a way that their surfaces are inmechanical contact so as to form additional chambers (11).
 15. A rotaryscrew machine according claim 1, characterized in that said female orderof symmetry Nf is equal to Nm−1.
 16. A rotary screw machine according toclaim 1, characterized in that said female order of symmetry Nf is equalto Nm+1.
 17. A rotary screw machine according to claim 1, characterizedin that said male and female surfaces can degenerate into cylindricalsurfaces.
 18. A method of transforming a motion in a volume screwmachine, the method comprising: (a) creating an interconnected motion ofscrew conjugated elements in the form of male and female members andlinks of synchronizing coupling with the help of converted positiveflows of mechanical energy and working substance energy in workingchambers of said volume screw machine; (b) driving one of male or femalemember into a planetary motion with two degrees of freedom of mechanicalrotation one of which being an independent degree of freedom relative tothe fixed central axis of the other member; and (c) transmitting saidpositive energy flows of conversion through an independent degree offreedom of mechanical rotation of said machine.
 19. The method accordingto claim 18, further comprising creating a differentially connectedmotion of male and female members and links of synchronizing couplingwith a second independent degree of freedom of a rotary motion and thetransmission of the positive energy flow of conversion in the form ofthe two flows through the two independent degrees of freedom.
 20. Themethod according to claim 18, in which the third, at least one dependentdegree of freedom of rotary motion, can be created in the process oftransforming a motion of male and female members and links ofsynchronizing coupling, and a part of positive energy flow of conversioninside said machine, is used in transforming a motion through anadditional dependent degree of freedom of mechanical rotation of saidmachine with decreasing the number of independent degrees of freedom perunity.
 21. The method according to claim 18, in which the angularvelocities of said members are determined according to the expression:k ₁ω₁ +k ₂ω₂+ω₃=O, where: ω₁,ω₂ represent the angular speed of the saidconjugated elements about their axis; ω₃ represents the angular speed ofthe link of synchronizing coupling; k₁,k₂ represent the constantcoupling coefficients; herewith, values of angular velocities ofrotation of conjugated elements are defined from expression:(z−1)ω₁ −zω ₂+ω₀=0, where: ω₁ represents is the angular speed of themember around its axis, enveloping surface of which has the form ofcurvilinear surface; ω₂ represents the angular speed of rotation of themember around its axis, enveloping surface of which has a shape of inneror outer envelope of a family of surfaces, formed with the saidcurvilinear surface; ω₀ represents the angular speed of the orbitalrevolution of the axis of the member executing planetary motion; zrepresents an integer, z>1.
 22. The method according to claim 18, inwhich any two of the three rotations can be synchronized between oneanother, namely, the rotation of one of the conjugated elements abouttheir fixed axis, the revolution of an axis of the member performing aplanetary motion with the link of synchronizing coupling and theswiveling of the member with a movable axis.